In Nouredine Zettili's QM book, Hilbert space $H$ is said to be separable when: There exists a Cauchy sequence $\psi_n \in H$ ($n=1,2,\ldots)$ such that for every $\psi$ of $H$ and $\varepsilon > 0,$ there exists at least one $\psi_n$ of the sequence for which $||\psi - \psi_n || < \varepsilon.$

But in Kreyszig's Functional Analysis book, separable is defined to be: A metric space $X$ is said to be separable if it has a countable subset which is dense in $X$.

I tried to look for the separability of of Hilbert space in Kreyszig's but couldn't find anything similar to Zettili's. My question then is, that are the above mentioned two definitions of separable space equivalent?